Our best friend's 92 y o mother is in a nursing home (X ~3 years). She has dementia and has suffered many falls and fractures over the years but is able to pivot into a chair. Her son died yesterday in a hospital (was there 10 months except a for a few two week or less unsuccessful stays at home. His mom rarely remembered he was even in the hospital though she was retold (honestly/briefly) whenever she asked where he was. I think I've read that it's recommended a parent or spouse be told once of their loved one's passing. Should this be before the funeral? I am capable of transporting her to the funeral home, or Mass if need be if this is what the family would want.
Three years later, his mother lingered for several months with pancreatic cancer. We went to her one time to see her just days before she died. When he saw her, he didn't want to be in the same room. He became extremely agitated and demanded we "go now." I believe he knew she was dying, even though by that time he could barely talk. She understood, but one of her wishes was to see him before she died and I honored that. She seemed glad we had come. I took leave of his sister and his mother's new husband (of less than a year) and told them we would not be attending the funeral. I offered my condolences, asked if they needed any help, and went back to our home town. She died two days later. With each of these times, my husband seemed to realize what had happened.
These are just examples of the reactions of one person at different stages of dementia. You will have to decide if your loved one is capable of understanding and behaving appropriately.
My mom was in the hospital with end stage CHF, when my son was diagnosed with Leukemia. I did not tell her. She did not need to deal with her grandson's illness when she was so ill herself. I do not regret that decision.
My job for my dad with dementia is to answer his questions as honestly as I can without creating hurt and more confusion. I try to creatively change the subject and he forgets what we were talking about.
Ultimately it is the family's decision, but certainly you can offer help in whatever they decide. You seem to be a caring friend. I think it would be a very nice gesture to offer to take care of mom if they decide to bring her to the funeral.
My opinion is that they should be told of their loved one's passing - but only if you think they can handle it without the pain and sadness being experienced repeatedly because they forgot that the person had passed and had to be told again. There's no reason to subject them to that repeatedly if it can be prevented. Yes, they have a right to know their loved one has passed, but if it is only going to cause more pain and confusion for them, then the benefit of their knowing needs to be weighed against that.
I saw the mother of a good friend of mine attend a formal church funeral for her husband/the friend's father, when the mother was already at a fairly advanced stage of dementia. I don't know if she'd been "dosed" beforehand at her NH but she certainly didn't behave in a way that upset or embarrassed or even mildly discomfited anyone, and naturally enough she didn't seem to be distressed herself - I wasn't sure she had any idea of what was going on.
If you're confident of getting this lady to and from the service safely, and the family would like her to be there (if only out of propriety and respect for the deceased), then it's doable. As to whether it could be of any benefit at all to the mother herself, who can say? I generally err on the side of inclusion and truth-telling, but leave the decision to those who know her best.
As for going to the funeral I think that would depend upon how far her dementia has progressed. Is she able to be transferred to and from a car? To and from a church pew? Does she get agitated? A long Mass and maybe a reception following might be too much for her. But this would be her family's decision.